Finite Group Databasebeta

← #124609

Subgroups — #124609

Order 512.

602 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#395C211yesmaximal, characteristic
2#8701C211yesmaximal, characteristic
3#8560C211yesmaximal, characteristic
4#7131C212yesmaximal
5#7123C212yesmaximal
6C22×C4×Q8C411yescharacteristic
7C2×#395C411yescharacteristic
8C2×C4×C4⋊C4—21no—
9C2×C4×C4⋊C4C2211yescharacteristic
10#1817—21no—
11#1797—21no—
12C2×#324—21no—
13#1446C2211yescharacteristic
14#1444—21no—
15#1444C2211yescharacteristic
16#1441—21no—
17#1441C411yescharacteristic
18C4×#95—21no—
19C4×#95C411yescharacteristic
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C23×Q8C2×C411yescharacteristic
24C2×C4×Q8—44no—
25C2×C4×Q8—41no—
26C2×C4×Q8—41no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C42—21no—
34C22×C42—21no—
35C22×C42C2×C411yescharacteristic
36C22×C42D411yescharacteristic
37C22×C42D411yescharacteristic
38C2×C4⋊C8—41no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41#395—22no—
42#395C814yescomplemented
43C4×C4⋊C4—41no—
44C4×C4⋊C4—41no—
45C4×C4⋊C4—44no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95D411yescharacteristic
56C2×#95C2311yescharacteristic
57#336—41no—
58#336—41no—
59#336—41no—
60#324—42no—
61C22×Q8—21no—
62C22×Q8—42no—
63C22×Q8—42no—
64C22×Q8—41no—
65C22×Q8—41no—
66C22×Q8C2×C812yes—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C22×C8—42no—
79C22×C8—42no—
80C4×Q8—44no—
81C4×Q8—44no—
82C4×Q8—44no—
83C4×Q8—44no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4C2×C812yes—
103C2×C4⋊C4M1612yes—
104C2×C42—24no—
105C2×C42—24no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—22no—
112C2×C42—21no—
113C2×C42—41no—
114C2×C42—42no—
115C2×C42—42no—
116C2×C42—42no—
117C2×C42—41no—
118C2×C42—41no—
119C2×C42—42no—
120C2×C42—42no—
121C4⋊C8—82no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126#95—24no—
127#95—21no—
128#95—41no—
129#95—41no—
130#95—41no—
131#95—42no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95—42no—
137#95C2×C812yes—
138C24C2×C22⋊C411yescharacteristic
139C2×Q8—44no—
140C2×Q8—44no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—48no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—42no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4C22⋊C812yes—
208C22×C4C22⋊C812yes—
209C22×C4#9911yescharacteristic
210C22×C4#9911yescharacteristic
211C22×C4C22×C811yescharacteristic
212C22×C4C2×M1611yescharacteristic
213C22×C4AGL(1,ℤ/8)11yescharacteristic
214C22×C4#13711yescharacteristic
215C22×C4#13712yes—
216C2×C8—82no—
217C2×C8—82no—
218C2×C8—82no—
219C2×C8—82no—
220C2×C8—82no—
221C2×C8—82no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—48no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—48no—
231C4⋊C4—42no—
232C42—42no—
233C42—42no—
234C42—48no—
235C42—44no—
236C42—42no—
237C42—42no—
238C42—44no—
239C42—42no—
240C42—81no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23C2×C22⋊C811yescharacteristic
246C23C2×#9911yescharacteristic
247C23#41311yescharacteristic
248C23#41711yescharacteristic
249C23#41911yescharacteristic
250C23#42111yescharacteristic
251C23#42111yescharacteristic
252Q8—44no—
253Q8—44no—
254Q8—44no—
255Q8—44no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4AΣL(1,𝔽16)12yes—
327C2×C4#35612yes—
328C8—162no—
329C8—162no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22#145911yescharacteristic
343C22#147711yescharacteristic
344C22#147911yescharacteristic
345C22#151111yescharacteristic
346C22#151311yescharacteristic
347C22#153511yescharacteristic
348C22#155911yescharacteristic
349C22C2×AΣL(1,𝔽16)11yescharacteristic
350C22#212011yescharacteristic
351C22#212611yescharacteristic
352C22C2×#35611yescharacteristic
353C4—22no—
354C4—22no—
355C4—44no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—42no—
360C4—44no—
361C4—41no—
362C4—41no—
363C4—42no—
364C4—82no—
365C4—81no—
366C4—82no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2—21no—
371C2#760011yescharacteristic
372C2#1147111yescharacteristic
373C2#1147811yescharacteristic
374C2#1181011yescharacteristic
375C2#1181511yescharacteristic
376C2#1182011yescharacteristic
377C2#1183111yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
42. #395C8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C4×#3952
#87012
#85602
#71312
#71232